Problem: Integrate. $\int\left(-2e^x-\dfrac6x \right)dx=\,?$ Choose 1 answer: Choose 1 answer: (Choice A) A $-2e^x-6\ln|x|+C$ (Choice B) B $e^{-2x}-6\ln(x)+C$ (Choice C) C $-2e^x-6\ln(x)+C$ (Choice D) D $e^{-2x}-6\ln|x|+C$
Solution: We can integrate using the following formulas for the indefinite integrals of $e^x$ and $\dfrac1x$ : $\begin{aligned} &\int e^x\,dx=e^x+C \\\\ &\int \dfrac1x\,dx=\ln|x|+C \end{aligned}$ $\begin{aligned} &\phantom{=}\int\left(-2e^x-\dfrac6x \right)dx \\\\ &=-2\int e^x\,dx-6\int\dfrac1x \,dx \\\\ &=-2e^x-6\ln|x|+C \end{aligned}$